Counting isomorphism classes of elliptic curves over $\mathbb{F}_q(t)$

Abstract

We determine the precise number of isomorphism classes of elliptic curves over $\mathbb{F}_q(t)$ with $\text{char}(\mathbb{F}_q) = 3,2$. The key idea is to obtain the exact unweighted number of rational points on the classifying stacks $\mathcal{B} Q_{12}$, $\mathcal{B} Q_{24}$ and $\mathcal{B} Z$, where $Q_{12}$ and $Q_{24}$ denote the dicyclic groups of orders 12 and 24, respectively, and $Z$ denotes the non-reduced group scheme of order 2. This computation, inspired by the classical work of [de Jong] and performed via motivic height zeta functions of height moduli spaces constructed in [Bejleri-Park-Satriano], establishes a complete determination of the total number of isomorphism classes of rational points on $\overline{\mathcal{M}}_{1,1}$ over any rational function field $k(t)$ with perfect residue field $\text{char}(k) \ge 0$.